What are non-held-out data or non-held-out classes?
Held-out simply means "not included" particularly in the sense of:

This part of the data was not included in this specific training run.

Depending on the context of all of these text non-held-out data/classes means the data that actually was included in a particular modeling exercise.

Consider this excerpt from your first example:


  For instance, Owen and Perry (2009) show a method for holding out data, computing SVD on the non-held-out data, and selecting k so as to minimize the reconstruction error between the held-out data and its SVD approximation.


It actually means:


  For instance, Owen and Perry (2009) show a method for excluding data, computing SVD on the remaining data, and selecting k so as to minimize the reconstruction error between the excluded data and its SVD approximation.


So it simply talks about a particular way of train-test-validation splitting the data.